### 21.4 Examples of Usage ¶

The following can be used to solve a linear system `A*x = b`

using the pivoted LU factorization:

[L, U, P] = lu (A); ## now L*U = P*A
x = U \ (L \ P) * b;

This is one way to normalize columns of a matrix `X` to unit norm:

s = norm (X, "columns");
X /= diag (s);

The same can also be accomplished with broadcasting (see Broadcasting):

s = norm (X, "columns");
X ./= s;

The following expression is a way to efficiently calculate the sign of a
permutation, given by a permutation vector `p`. It will also work
in earlier versions of Octave, but slowly.

det (eye (length (p))(p, :))

Finally, here’s how to solve a linear system `A*x = b`

with Tikhonov regularization (ridge regression) using SVD (a skeleton
only):

m = rows (A); n = columns (A);
[U, S, V] = svd (A);
## determine the regularization factor alpha
## alpha = ...
## transform to orthogonal basis
b = U'*b;
## Use the standard formula, replacing A with S.
## S is diagonal, so the following will be very fast and accurate.
x = (S'*S + alpha^2 * eye (n)) \ (S' * b);
## transform to solution basis
x = V*x;